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2017 Detection of low dimensionality and data denoising via set estimation techniques
Catherine Aaron, Alejandro Cholaquidis, Antonio Cuevas
Electron. J. Statist. 11(2): 4596-4628 (2017). DOI: 10.1214/17-EJS1370


This work is closely related to the theories of set estimation and manifold estimation. Our object of interest is a, possibly lower-dimensional, compact set $S\subset{\mathbb{R}}^{d}$. The general aim is to identify (via stochastic procedures) some qualitative or quantitative features of $S$, of geometric or topological character. The available information is just a random sample of points drawn on $S$. The term “to identify” means here to achieve a correct answer almost surely (a.s.) when the sample size tends to infinity. More specifically the paper aims at giving some partial answers to the following questions: is $S$ full dimensional? Is $S$ “close to a lower dimensional set” $\mathcal{M}$? If so, can we estimate $\mathcal{M}$ or some functionals of $\mathcal{M}$ (in particular, the Minkowski content of $\mathcal{M}$)? As an important auxiliary tool in the answers of these questions, a denoising procedure is proposed in order to partially remove the noise in the original data. The theoretical results are complemented with some simulations and graphical illustrations.


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Catherine Aaron. Alejandro Cholaquidis. Antonio Cuevas. "Detection of low dimensionality and data denoising via set estimation techniques." Electron. J. Statist. 11 (2) 4596 - 4628, 2017.


Received: 1 March 2017; Published: 2017
First available in Project Euclid: 18 November 2017

zbMATH: 1383.62078
MathSciNet: MR3724969
Digital Object Identifier: 10.1214/17-EJS1370

Primary: 62G05
Secondary: 60D05


Vol.11 • No. 2 • 2017
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